Solutions to problems with imprecise data—An engineering perspective to generalized uncertainty models S. Pannier n , M. Waurick, W. Graf, M. Kaliske Institute for Structural Analysis, Technische Universit¨ at Dresden, 01062 Dresden, Germany article info Article history: Received 9 August 2011 Received in revised form 21 May 2012 Accepted 2 August 2012 Available online 28 September 2012 Keywords: Fuzzy probability based fuzzy randomness Generalized uncertainty models Fuzzy randomness Fuzzy probability Imprecise data abstract The present paper is about data corrupted by both aleatoric and epistemic uncertainty. A unification of randomness, which represents aleatoric uncertainty, and fuzziness, which represents epistemic uncertainty, is dicussed in detail. As a result, the main uncertainty characteristics, i.e., variability, incompleteness and imprecision, can be described. With a focus on engineering problems the aim is to bridge the imprecision of data to the decision making process. Suitable fields of applications are highlighted; remarks on the numerical treatment are given. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction Uncertainty in input parameters arises from both the natural variability in parameters, which cannot be reduced without unreasonable expense, and incompleteness and imprecision of available information due to lack of knowledge, subjective assessments, incapability of exact measurements, etc. Generally, the information on input parameters is represented by data sets, which are gained conventionally with quasi objective specifications like measurements, experiments, and so forth. Since plenty of information cannot be generated in this manner, an increased awareness about the importance of subjective evaluations and expert specifications arose in the recent past [30]. In engineering applications, available data are afflicted different kinds of uncertainty [16]. These are, among other things, variability, incompleteness and imprecision (denoted respectively as physical uncertainty, statistical uncertainty and perceptional uncertainty). Usually, incompleteness is an umbrella term, which also incorporates imprecision [24]. Here, an insufficient amount of (objective) data is referred to as incompleteness. The inability of observing physical experiments in a precise manner is indicated as imprecision. The diversity of sources of such uncertainties impedes the application of a single uncertainty model. Consequently, generalized formulations of the respective uncertainty models are reasonable. Generally, uncertainty can be classified into aleatoric and epistemic uncertainty. Aleatoric uncertainty implies chance, variability and randomness and is mostly modeled with the aid of stochastic quantities. Epistemic uncertainty implies, among other things, imprecision and incompleteness. Since stochastic methods are powerful tools, much effort concentrates on applying them to model epistemic uncertainty as well [12,13,16]. In contrast, it is argued that alternative Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/ymssp Mechanical Systems and Signal Processing 0888-3270/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ymssp.2012.08.002 n Corresponding author. Tel.: þ49 351 46334406; fax: þ49 351 46337086. E-mail addresses: stephan.pannier@tu-dresden.de (S. Pannier), marcus.waurick@tu-dresden.de (M. Waurick), wolfgang.graf@tu-dresden.de (W. Graf), michael.kaliske@tu-dresden.de (M. Kaliske). Mechanical Systems and Signal Processing 37 (2013) 105–120